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Let $$O(1,1)=\{A\in GL_n(F):A^tJA=J\}$$ where $$J= \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix}$$

My question is; Let $$T=\{A\in O(1,1): A=\begin{pmatrix} cosht & sinht \\ sinht & cosht \\ \end{pmatrix} ,t\in \Re\}$$ Then, is $T$ a normal subgroup of $O(1,1)$?

This is puzzling since we can do coset decomposition on $O(1,1)$ modulo $T$. I'd really appreciate any hint for this.

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    $\begingroup$ 1. What field do you consider $O(1,1)$ over? 2. Why do you say that this is puzzling? Do you think that $T$ should not be a normal subgroup? I don't understand how the first and second parts of your question go together. $\endgroup$ – KReiser Dec 6 '16 at 6:43

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